Screening Test for a Warehouse Defect
A packing line produces components of which 3% are defective. An automated vision check flags a component as defective with probability 0.92 if it truly is defective, and with probability 0.05 if it is not. Flagged components are pulled for manual inspection; the manual inspector correctly identifies a truly defective item 95% of the time and wrongly rejects a good item 2% of the time. The line supervisor sees that 40 components were flagged last shift out of 1000 produced and wants to know how many actual defects to expect, and whether the two-stage system is worth its cost.
1. Compute the probability that a component is truly defective given that it was flagged by the vision check, and the expected number of true defects among 1000 components.
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Let D = defective, F = flagged. P(D) = 0.03, P(F|D) = 0.92, P(F|not D) = 0.05. P(F) = 0.92(0.03) + 0.05(0.97) = 0.0276 + 0.0485 = 0.0761. P(D|F) = 0.0276 / 0.0761 ≈ 0.3627, so about 36% of flagged items are truly defective. Expected defects in 1000: 0.03 × 1000 = 30. Expected flags: 0.0761 × 1000 ≈ 76. The observed 40 flags is well below expectation, which is a warning sign that either the defect rate or the flag rates are mis-specified, or the shift data are incomplete.
2. The supervisor wants the probability that a component is finally rejected by the full two-stage system. Compute it and comment on whether the second stage adds value.
Read the worked answer
A component is finally rejected if flagged and then rejected by the inspector. If truly defective and flagged: 0.0276; inspector rejects with 0.95, contribution 0.0276 × 0.95 = 0.02622. If good and flagged: 0.0485; inspector wrongly rejects with 0.02, contribution 0.0485 × 0.02 = 0.00097. Total rejection probability ≈ 0.02622 + 0.00097 = 0.02719, so about 2.7% of all components are rejected. Precision of the final rejection: 0.02622 / 0.02719 ≈ 0.964, so about 96% of rejected items are genuinely defective, versus 36% after stage one alone. The second stage substantially improves precision and is worth its cost, though it also means roughly 1 in 1000 good components is wrongly rejected.